Abstract and Applied Analysis

Volume 2013 (2013), Article ID 908168, 8 pages

http://dx.doi.org/10.1155/2013/908168

Research Article

Solution and Stability of Euler-Lagrange-Rassias Quartic Functional Equations in Various Quasinormed Spaces

Heejeong Koh and Dongseung Kang

Department of Mathematical Education, Dankook University, 126 Jukjeon, Suji, Yongin, Gyeonggi 448-701, Republic of Korea

Received 9 July 2013; Accepted 3 September 2013

Academic Editor: Nicole Brillouet-Belluot

Copyright © 2013 Heejeong Koh and Dongseung Kang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We obtain the general solution of Euler-Lagrange-Rassias quartic functional equation of the following . We also prove the Hyers-Ulam-Rassias stability in various quasinormed spaces when .

1. Introduction

One of the interesting questions concerning the stability problems of functional equations is as follows: when is it true that a mapping satisfying a functional equation approximately must be close to the solution of the given functional equation? Such an idea was suggested in 1940 by Ulam [1] as follows. Let be a group and let be a metric group with the metric . Given , does there exist a such that if a function satisfies the inequality for all then there is a homomorphism with for all ? In other words, we are looking for situations when the homomorphisms are stable; that is, if a mapping is almost a homomorphism, then there exists a true homomorphism near it. In 1941, Hyers [2] considered the case of approximately additive mappings in Banach spaces and satisfying the well-known weak Hyers inequality controlled by a positive constant. The famous Hyers stability result that appeared in [2] was generalized in the stability involving a sum of powers of norms by Aoki [3]. In 1978, Rassias [4] provided a generalization of Hyers Theorem which allows the Cauchy difference to be unbounded. During the last decades, stability problems of various functional equations have been extensively studied and generalized by a number of authors [5–10]. In particular, Rassias [11] introduced the Euler-Lagrange type quadratic functional equation

for fixed reals , with , . Also, Jun and Kim [12] proved the Hyers-Ulam-Rassias stability of a Euler-Lagrange type cubic mapping as follows:

where , , , for all . Several Euler-Lagrange type functional equations have been investigated by numerous mathematicians; c.f. for example, [13–15].

And Rassias [16] investigated stability properties of the following quartic functional equation:

It is easy to see that is a solution of (3) by virtue of the identity

For this reason, (3) is called a quartic functional equation. Also, Chung and Sahoo [17] determined the general solution of (3) without assuming any regularity conditions on the unknown function. In fact, they proved that the function is a solution of (3) if and only if , where the function is symmetric and additive in each variable. Lee and Chung [18] introduced a quartic functional equation as follows:

for fixed integer with , .

In this paper, we consider the following a generalized quartic functional equation:

for fixed integers and such that , , , for all . In fact, the generalized quartic functional equation (6) is following from the spirit of the pioneering Euler-Lagrange quartic functional equation (3) as well as Euler-Lagrange quadratic functional equation (1) introduced by Rassias: see [16] and [11], respectively. For the same reason as (1), (2), and (3), we call (6) a Euler-Lagrange-Rassias quartic functional equation. First of all, we obtain the general solution of Euler-Lagrange-Rassias quartic functional equation. To prove the stability problem for the Euler-Lagrange-Rassias quartic functional equation on various quasi-normed spaces, we may consider the following:

for fixed integer with , , for all .

We will use the following definitions to prove Hyers-Ulam-Rassias stability for the Euler-Lagrange-Rassias quartic functional equation in the quasi--normed and quasi fuzzy -normed spaces. Let be a real number with and let be either or .

Definition 1. Let be a linear space over a field . A quasi -norm   is a real-valued function on satisfying the following statements:

(1) for all and if and only if ,(2) for all and all ,(3)there is a constant such that for all .

The pair is called a quasi- -normed space if is a quasi--norm on . The smallest possible is called the modulus of concavity of . A quasi- -Banach space is a complete quasi--normed space.

A quasi -norm is called a -norm ( if (3) takes the form for all . In this case, a quasi -Banach space is called a -Banach space; see [19, 20].

In 1984, Katsaras [21] and Wu and Fang [22] independently introduced a notion of a fuzzy norm and they gave the generalization of the Kolmogoroff normalized theorem for a fuzzy topological linear space. Since then, some mathematicians have defined fuzzy metrics and norms on a linear space from various points of view; see [23–27]. In 2003, Bag and Samanta [23] modified the definition of Cheng and Mordeson [28]. Bag and Samanta [23] introduced the following definition of fuzzy normed spaces. The notion of fuzzy stability of functional equations was given in the paper [29].

Definition 2. Let be a real vector space. A function is called a fuzzy norm on if for all and all

) for ;() if and only if for all ;() if ;();() is a nondecreasing function of and ;() for ,   is continuous on .

The pair () is called a fuzzy normed vector space.

Mirmostafaee [30] introduced a notion for a quasi fuzzy -normed space as follows.

Definition 3. By a quasi fuzzy norm, one means a real vector space , with a fuzzy subset of and some such that all axioms of fuzzy normed space in Definition 2 except () and

()

hold.

A quasi fuzzy normed space () which satisfies

()

for some , is called a quasi fuzzy -norm.

Definition 4. Let be a real vector space. A quasi fuzzy -norm is called a quasi fuzzy  -norm on if () in Definition 2 takes the form.

Example 5. Let be a real normed space. Define

where . Then is a quasi fuzzy -normed space.

Note that when , we call the quasi fuzzy -norm a quasi fuzzy -norm.

Definition 6. Let () be a quasi fuzzy -normed vector space. A sequence in is said to be convergent or converge if there exists an such that for all . In this case, is called the limit of the sequence and one denotes it by .

Definition 7. Let () be a quasi fuzzy -normed vector space. A sequence in is called Cauchy if for each and each there exists an such that, for all and all integer , one has .

It is well known that every convergent sequence in a quasi fuzzy -normed vector space is Cauchy. If each Cauchy sequence is convergent, then the quasi fuzzy -normed space is said to be quasi fuzzy complete and the quasi fuzzy -normed vector space is called a quasi fuzzy Banach space.

2. Euler-Lagrange-Rassias Quartic Functional Equations

Let , be real vector spaces. In this section, we will investigate that the functional equation (3) is equivalent to the presented functional equation (6).

Lemma 8. A mapping satisfies the functional equation (3) if and only if satisfies

for all .

Proof. It follows from [31, 32].

Theorem 9. A mapping satisfies the functional equation (3) if and only if satisfies the functional equation (7).

Proof. It is easy to verify that by letting in (3). We will show this induction on . Lemma 8 implies that it is true when , and we may assume it holds for all . Now, letting and in (3), we have

for all . After the switching and in the previous equation (13),

for all . Adding two equations (13) and (14), we have

for all . The induction steps imply that

for all . Hence we have

for all , as desired.

Note that by letting in (7).

Lemma 10. A mapping satisfies the functional equation (7) if and only if satisfies the functional equation (6).

Proof. It is easy to show that and by putting and in (7), respectively. By letting in (7), we have

Also, switching and in the above equation and then adding two equations, we get

Then (7) implies that

Corollary 11. A mapping satisfies the functional equation (3) if and only if satisfies the functional equation (6).

3. Stability in Quasi-β-Normed Spaces

Throughout this section, let be a quasi--normed space and let be a quasi -Banach space with a quasi -norm . Let be the modulus of concavity of . We will investigate the Hyers-Ulam-Rassias stability problem for the functional equation (7). For a given mapping and all fixed integers with , , let

for and in .

Theorem 12. Suppose that there exists a mapping for which a mapping satisfies and

and the series converges for all . Then there exists a unique Euler-Lagrange-Rassias quartic mapping which satisfies (7) and the inequality

for all .

Proof. By letting in (22) and , we have

that is,

for all . For any positive integer , we have

for all . For any positive integers and with ,

for all . By letting , we have

for all and . Since the right-hand side of the previous inequality tends to 0 as , is a Cauchy sequence in the quasi -Banach space . Thus we may define

for all . Hence we have the inequality (23). Since , replacing and by and , respectively, and dividing by in (22), we have

for all . By taking , the definition of implies that satisfies (7) for all ; that is, is the Euler-Lagrange-Rassias quartic mapping. It is left to show that the quadratic mapping is unique. Assume that there exists satisfying (7) and (23). Then

for all . By letting , we immediately have the uniqueness of .

Theorem 13. Suppose that there exists a mapping for which a mapping satisfies and

and the series converges for all . Then there exists a unique Euler-Lagrange-Rassias quartic mapping which satisfies (7) and the inequality

for all .

Proof. If is replaced by in inequality (25), we have

for all . The remains of the proof follow from the proof of Theorem 12.

4. Stability in Quasi Fuzzy β-Normed Spaces

Let us fix some notations which will be used throughout this section. We assume is a vector space and () is a quasi fuzzy -Banach space. We will prove the Hyers-Ulam-Rassias stability of the functional equation satisfying equation (7) in quasi fuzzy -Banach space.

Theorem 14. Let be a function such that for some

for all and all . Let be a mapping satisfying and

for all and all .

Then exists for each and defines a unique Euler-Lagrange-Rassias quartic mapping such that

for all and all .

Proof. Let in inequality (36). Since , we have

for all and all . Replacing by in inequality (38),

that is,

for all , and . Since ,

By letting in the previous inequality,

for all and all . Hence we get

for all and all . Letting in the previous inequality, we have

that is,

for all and all . Letting , we have

for all ,   and . Hence is a Cauchy sequence in the quasi fuzzy -Banach space . Thus, we may define

for all . Hence inequality (43) implies that

for large enough and all . Taking the limit as and using (), we have

for all . Hence it satisfies inequality (37). Now letting and in (36),

for all and all . This implies that

for all and all . Since , we may conclude that the mapping satisfies (7); that is, is the Euler-Lagrange-Rassias quartic mapping. It is left to show that the quartic mapping is unique. Assume there is another satisfying (7) and inequality (37). For each , clearly and for all .

for all and . Since , we have . Hence ; that is, the mapping is unique, as desired.

Theorem 15. Let be a function such that for some

for all and all . Let be a mapping satisfying and

for all and all .

Then exists for each and defines a unique Euler-Lagrange-Rassias quartic mapping such that

for all and all .

Proof. The techniques are completely similar to the proof of Theorem 14. Hence we present some key idea of this proof. Let in inequality (54). Since , we have

for all and all . Replacing by in inequality (56), we have

or

for all and all . For positive integers and ,

for all and . Hence we may conclude that is a Cauchy sequence in the quasi fuzzy -Banach space . Thus we may define

for all . Also, for any positive integer , we get

for all and all . This implies inequality (55).

Acknowledgment

The present research was conducted by the research fund of Dankook University in 2013.

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